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In algebraic geometry, a morphism between schemes is said to be smooth if (i) it is locally of finite presentation (ii) it is flat, and (iii) for every geometric point the fiber is regular. (iii) means that each geometric fiber of f is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If S is the spectrum of an algebraically closed field and f is of finite type, then one recovers the definition of a nonsingular variety. There are many equivalent definitions of a smooth morphism. Let be locally of finite presentation. Then the following are equivalent. f is smooth. f is formally smooth (see below). f is flat and the sheaf of relative differentials is locally free of rank equal to the relative dimension of . For any , there exists a neighborhood of x and a neighborhood of such that and the ideal generated by the m-by-m minors of is B. Locally, f factors into where g is étale. Locally, f factors into where g is étale. A morphism of finite type is étale if and only if it is smooth and quasi-finite. A smooth morphism is stable under base change and composition. A smooth morphism is universally locally acyclic. Smooth morphisms are supposed to geometrically correspond to smooth submersions in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by Ehresmann's theorem). Let be the morphism of schemes It is smooth because of the Jacobian condition: the Jacobian matrix vanishes at the points which has an empty intersection with the polynomial, since which are both non-zero. Given a smooth scheme the projection morphism is smooth. Every vector bundle over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of over is the weighted projective space minus a point sending Notice that the direct sum bundles can be constructed using the fiber product Recall that a field extension is called separable iff given a presentation we have that .

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Glossary of algebraic geometry

This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

Smooth scheme

In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n.

Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties.

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